Section Number:
4.4 / Topics: The First Fundamental Theorem of Calculus / Day: 50
The First Fundamental Theorem Of Calculus
Example 1: Evaluate the following definite integrals.
a. b. c.
Example 2: Evaluate the following definite integral using the definition of absolute value.
Example 3: Use the Fundamental Theorem of Calculus to find the area bounded by the graph of , the x-axis, and the vertical lines and .
The Mean Value Theorem for Integrals
Demonstration:
The Average Value of a Function
Example 4: Find the average value of on the interval .
Example 5: Oil is leaking from a storage tank at the rate ofgallons/day.
a. Graph the curve on the coordinate plane to the
right.
b. How much oil has leaked out by the end of the
fifth day?
c. How much oil has leaked out between the fifth
and tenth day? (i.e. on )?
Why is this answer less than that in part b?
d. What is the average daily loss of oil that is leaked out over these 20 days?
AP CALCULUS - AB / MR. RECORDSection Number:
4.4 / Topics: The Second Fundamental Theorem of Calculus / Day: 51
The Second Fundamental Theorem Of Calculus
The Definite Integral as a NumberThe Definite Integral as a function of x
Example 6: Evaluate the following.
a. b.
Example 7: Evaluate the following.
a. b.
Example 8: A radio-controlled experimental vehicle, an RC-XD, is tested on a straight track. It starts from rest, and its velocity v (meters per second) is recorded in the table below every 10 seconds.
t / 0 / 10 / 20 / 30 / 40 / 50 / 60v / 0 / 5 / 21 / 40 / 62 / 78 / 83
a. Using your TI-Nspire, find a cubic regression equation that models this data.
b. Use the Fundamental Theorem of Calculus to approximate the total distance traveled by the vehicle during
this test run.
Example 9: Consider a particle moving along the x-axis where x(t) is the position of the particle at time t (in terms of its distance from the origin), is its velocity, and is the distance the particle travels over time.
a. Complete the table below that gives the particles position at each of the 5 time intervals.
t / 0 / 1 / 2 / 3 / 4 / 5x
b. Find the total distance the particle travels in 5 units of time.
c. Why is there an absolute value in the expression .
d. Find the total displacement of the particle during these 5 units of time.
Example 10: Let where the graph of is below. Remember is the same thing as Thinks of as the rateof snowfall over a period of time. For instance at x = 1, snow if falling at a rate of 3 inches per hour, at x = 3, it is not snowing, and at x=4, snow is melting at a rateof 4 inches per hour.
a. Complete the chart. In the snow analogy, erpresents the accumulation of snow over time.
x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 6.5 / 7 / 8b. Now let’s consider If we take the derivative of an integral, what would you expect to
happen? ______.
So, is the same thing as ______.
Knowing that, let’s complete the chart.
x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 6.5 / 7 / 8c. On what subintervals of is F is increasing? Decreasing?
d. Where in the interval does Fachieve its minimum and maximum value? What are those values?
e. Find the concavity of F and any inflection points. Justify your answer.
f. Sketch a rough graph of .